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Against ‘Interpretation’: Quantum Mechanics Beyond Syntax and Semantics

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Abstract

The question “what is an interpretation?” is often intertwined with the perhaps even harder question “what is a scientific theory?”. Given this proximity, we try to clarify the first question to acquire some ground for the latter. The quarrel between the syntactic and semantic conceptions of scientific theories occupied a large part of the scenario of the philosophy of science in the 20th century. For many authors, one of the two currents needed to be victorious. We endorse that such debate, at least in the terms commonly expressed, can be misleading. We argue that the traditional notion of “interpretation” within the syntax/semantic debate is not the same as that of the debate concerning the interpretation of quantum mechanics. As much as the term is the same, the term “interpretation” as employed in quantum mechanics has its meaning beyond (pure) logic. Our main focus here lies on the formal aspects of the solutions to the measurement problem. There are many versions of quantum theory, many of them incompatible with each other. In order to encompass a wider variety of approaches to quantum theory, we propose a different one with an emphasis on pure formalism. This perspective has the intent of elucidating the role of each so-called “interpretation” of quantum mechanics, as well as the precise origin of the need to interpret it.

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Notes

  1. We will try to clear up confusion related to the terms “language” and “meaning”. Terms are the linguistic representations for certain quantities—for example, the term “mass” designates the physical quantity known as mass. Measuring, thus, is related to quantities, not their linguistic representations. In this sense, terms, of course, cannot be measured.

  2. Probably the first author to point out this problem was Maxwell (1962).

  3. It’s not our point here to discuss, in a profound way, how the traditional distinction can be if necessary, safeguarded [a more detailed discussion can be found at Silva (2013, pp. 144–151)]. D Lewis (1970) and Maxwell (1962) suggested how this can be done from a realist perspective and the recent developments in the structural realist view of theories depend/draw heavily on an adequate distinction between observational and theoretical terms.

  4. From this point forward, we will adopt the notation presented by Krause and Arenhart (2016), who call first-order structures as “order-1”, and high-order structures (e.g., with \(n>1\)) as “order-n”.

  5. Something that, again, Carnap was aware (cf. Carnap 1966, p. 193)

  6. (cf. Carnap 1959, p. 158) and Carnap (1966, p. 228).

  7. In this sense, Carnap is in partial disagreement with Quine’s holism: i.e., recognizing the continuum between theoretical and observational terms—and so only partially agreeing with Quine’s holism—but allowing a distinction to be justified (only) by practical purposes, suspending the judgment about its “nature”, that he would regard as a (suspicious) metaphysical commitment.

  8. For a detailed discussion see Silva (2020, pp. 87–95).

  9. It is noteworthy to mention that the terminology of “partial interpretation” is employed here in a non-standard way. It shall not to confused with its standard meaning: if every elementary statement in a theory T has a correspondent in its models, then it is considered to be a “full interpretation”; otherwise, it is a “partial interpretation” (cf. Haskell (1963, p. 48).

  10. We’ll say more about this in Sect. 4.

  11. It is not our goal to discuss the implications of this view for scientific realism; the reader interested in this subject may find a helpful discussion in Chakravartty (2001) and references therein.

  12. Although Chang and Keisler (1990) work only with order-1 structures, the idea can be generalized.

  13. This is also the case for non-standard semantics, such as Henkin semantics, see Henkin (1950) and Enderton (2015, Sect. 3). Nevertheless, we will stick with the standard case.

  14. For simplicity, from this point forward the description of these structures’ components shall be fixed as defined in the previous paragraph.

  15. See also Lutz (2015, Sect. 5).

  16. This issue is discussed in more detail in Sect. 6.

  17. Another way of stating such problem is relating “open” and “closed” systems—see Pessoa Junior (1997). This work, however, sticks to Maudlin’s taxonomy as it better relates to the literature here considered.

  18. In the sense of the word ‘interpretation’, as usually employed in QM.

  19. In his “Against ‘measurement”’, Bell (1990) put forth his famous critique of how the notion of “measurement” is ill-defined and even inadequate to the context of the foundations of QM. Our goal here is essentially to do the same, but with the term “interpretation”.

  20. For a critical summary of various formulations of QM, see Wightman (1976), Gudder (1979), Styer et al. (2002), and references therein.

  21. For those cases, see the references cited in Ćirković (2005, p. 821).

  22. For a brief analysis of seven thought experiments that show the differences in experimental results between collapse and no-collapse approaches versions of QM, see Ćirković (2005, pp. 823–834).

  23. An earlier version of \(\hbox {QM}^{bas}\) was presented in Arroyo (2020, Chap. 2).

  24. It is assumed, though, that this can be done—even if not here.

  25. See \({\mathcal {R}}\) ahead.

  26. Not to be confused with the “cosmic exile” mentioned in Sect. 4.

  27. There is some consensus concerning this mode of presentation of measurement results. For the sake of precision, it is essential to emphasize that this applies to measurements of the first kind. There are, however, measurements where this does not occur: where the eigenstate does not correspond to the eigenvalue. The position measurement satisfies the postulate presented, but, strictly, this does not apply to the measurement of energy.

  28. Recall that such a specific axiom is neutral regarding the question whether the states or the systems are prone to the branching process.

  29. Basically, this is the lead done by Niels Bohr (cf. Faye 2012, and references therein).

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Acknowledgements

We want to thank all the attendants of the VI International Workshop on Quantum Mechanics and Quantum Information: Identity and Individuality, held in Florianópolis, Brazil, 2019, in which part of the content of this paper was presented. In special, we also thank Adonai Sant’Anna, Christian de Ronde, David Wallace, Décio Krause, Eduardo Quirino, Jerzy Brzozowski, Jonas Arenhart, Kherian Gracher, Lauro Nunes Filho, Matheus Valente, Osvaldo Pessoa Jr., and Otávio Bueno for many fruitful discussions on related subjects, and for comments made on previous versions of this paper. We would also like to thank the anonymous reviewer for having raised several questions that allowed us to improve our article. Part of the content of this paper was also presented in chapters 1 and 2 of one of the authors’ doctoral thesis (RWA), which was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

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  1. Graduate Program in Philosophy (PPGFIL), Federal University of Santa Catarina (UFSC), Florianópolis, Santa Catarina, Brazil

    Raoni Wohnrath Arroyo

  2. Research Group on Logic and Foundations of Science (CNPq), International Network on Foundations of Quantum Mechanics and Quantum Information https://quantuminternationalnet.com/

    Raoni Wohnrath Arroyo

  3. Graduate Program in Philosophy (PPGF), University of Campinas (UNICAMP), Campinas, São Paulo, Brazil

    Gilson Olegario da Silva

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Arroyo, R.W., da Silva, G.O. Against ‘Interpretation’: Quantum Mechanics Beyond Syntax and Semantics. Axiomathes 32, 1243–1279 (2022). https://doi.org/10.1007/s10516-021-09579-y

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